Problem: The polynomial $P$ is graphed. ${1}$ ${2}$ ${\llap{-}1}$ ${\llap{-}2}$ ${\llap{-}3}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ $y$ $x$ $ P$ Graph of cubic polynomial P that passes through the points: negative 3, 2; negative 2, negative 4; and 2, 0. What is the remainder when $P(x)$ is divided by $(x+2)$ ?
Explanation: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x+2)$, which can be written as $(x-({-2}))$, is $P({-2})$. According to the graph, $P({-2})=-4$. ${1}$ ${2}$ ${\llap{-}1}$ ${\llap{-}2}$ ${\llap{-}3}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ $y$ $x$ $ P$ Same graph as above with point marked at negative 2, negative 4 In conclusion, the remainder when $P(x)$ is divided by $(x+2)$ is $-4$.